cp's OEIS Frontend

A382617 Numbers k such that k = m*(m^2 + 1) where m^2 + 1 is prime.

%I A382617 #15 Apr 15 2025 15:16:32
%S A382617 2,10,68,222,1010,2758,4112,8020,13848,17602,46692,64040,157518,
%T A382617 175672,287562,405298,592788,729090,830678,1331110,1561012,1728120,
%U A382617 1906748,2000502,2197130,2406238,3112282,3375150,3796572,4096160,4913170,5451952,5832180,6229688
%N A382617 Numbers k such that k = m*(m^2 + 1) where m^2 + 1 is prime.
%C A382617 Numbers k that are the nontrivial solutions to A070304 for primes p = m^2 + 1. Conjecture: for such primes p, there are exactly two solutions, k = 0 and the nontrivial k = m * p. These k correspond to the terms of this sequence.
%H A382617 Alois P. Heinz, <a href="/A382617/b382617.txt">Table of n, a(n) for n = 1..10000</a>
%H A382617 Jon Grantham and Hester Graves, <a href="https://arxiv.org/abs/2502.03513">Primes of the Form m^2 + 1 and Goldbach's "Other Other" Conjecture</a>, arXiv preprint arXiv:2502.03513 [math.NT], 2025.
%F A382617 a(n) = A005574(n)^3 + A005574(n).
%F A382617 a(n) = A005574(n)*A002496(n).
%e A382617 For a(1), m = 1, p = 1^2 + 1 = 2, and k = 1*2 = 2.
%e A382617 For a(2), m = 2, p = 2^2 + 1 = 5, and k = 2*5 = 10.
%e A382617 For a(3), m = 4, p = 4^2 + 1 = 17 and k = 4*17 = 68.
%p A382617 f:= m-> (p-> `if`(isprime(p), p*m, [][]))(m^2+1):
%p A382617 map(f, [$0..200])[];  # _Alois P. Heinz_, Mar 24 2025
%t A382617 Table[m*(m^2 + 1), {m, Select[Range[200], PrimeQ[#^2 + 1] &]}] (* _Amiram Eldar_, Mar 24 2025 *)
%o A382617 (Python)
%o A382617 from sympy import isprime
%o A382617 from itertools import count, islice
%o A382617 def agen(): # generator of terms
%o A382617     yield from (m*p for m in count(1) if isprime(p:=m**2+1))
%o A382617 print(list(islice(agen(), 34))) # _Michael S. Branicky_, Mar 24 2025
%o A382617 (PARI) apply(x->x*(x^2+1), select(x->isprime(x^2+1), [1..200])) \\ _Michel Marcus_, Mar 24 2025
%Y A382617 Cf. A002496, A005574, A070304.
%K A382617 nonn
%O A382617 1,1
%A A382617 _Steven Lee Benjamin_, Mar 31 2025